# `y = sin(root(3)(x)) + root(3)(sin(x))` Find the derivative of the function.

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Expert Answers

Borys Shumyatskiy | Certified Educator

Hello!

We have the composite functions, for such a functions

`[f(g(x))]' = f'(g(x))*g'(x).`

Also recall that `[sin(y)]' = cos(y)` and `[y^(1/3)]' = (1/3)*y^(-2/3).`

Let's start:

`y(x) = sin(x^(1/3)) + (sin(x))^(1/3)` .

`y'(x) = [sin(x^(1/3))]' + [(sin(x))^(1/3)]' =`

`cos(x^(1/3))*[x^(1/3)]' + (1/3)*(sin(x))^(-2/3)*[sin(x)]' =`

`(1/3)*x^(-2/3)*cos(x^(1/3)) + (1/3)*cos(x)*(sin(x))^(-2/3).`